Statistics of first-passage Brownian functionals

TL;DR

This paper derives exact and approximate distributions for first-passage functionals of Brownian motion, revealing tail behaviors, phase transitions, and large deviation principles, with applications to drifted and driftless cases.

Contribution

It provides exact solutions for the distribution of first-passage functionals for all n>-2 in the driftless case and develops a large-deviation framework with phase transition analysis for drifted cases.

Findings

Exact probability density for driftless case for all n>-2.

Tail behaviors: essential singular and power-law tails.

Identification of dynamical phase transitions in the rate function.

Abstract

We study the distribution of first-passage functionals ${\cal A}= \int_0^{t_f} x^n(t)\, dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=\text{Prob}.(\mathcal{A}=A)$. This probability density has an essential singular tail as $A\to 0$ and a power-law tail $\sim A^{-(n+3)/(n+2)}$ as $A\to \infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $A\to 0$ it predicts the same essential singular tail as in the driftless case. For $A\to \infty$ it predicts a stretched exponential tail $-\ln P_n(A|x_0)\sim A^{1/(n+1)}$ for all $n>0$. In the limit of large P\'eclet number $\text{Pe}= \mu x_0/(2D)\gg 1$, where $\mu$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-\ln P_n(A|x_0)\simeq\text{Pe}\, \Phi_n\left(z= A/\bar{A}\right)$, where $\bar{A}=x_0^{n+1}/{\mu(n+1)}$ is the mean value of $\mathcal{A}$. We compute the rate function $\Phi_n(z)$ analytically for all $n>-1$. For $n>0$ $\Phi_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $\mu<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $\mathcal{A}$ coincides with the distribution of $\mathcal{A}$ for $\mu>0$ with the same $|\mu|$.